Problem Analysis

The question asks for the number of unique right triangles PQR that can be constructed under specific conditions:

• The right angle is at vertex P.
• Side PR is parallel to the x-axis.
• All coordinates (x, y) of P, Q, and R must be integers.
• The coordinates must satisfy the inequalities: $-4 ≤ x ≤ 5$ and $6 ≤ y ≤ 16$.

Coordinate Constraints

First, determine the number of possible integer values for x and y coordinates:

• X-coordinates: Integers from -4 to 5 inclusive. The total number of possible x-values is $5 - (-4) + 1 = 10$.
• Y-coordinates: Integers from 6 to 16 inclusive. The total number of possible y-values is $16 - 6 + 1 = 11$.

Triangle Properties and Point Relationships

Given that the right angle is at P and PR is parallel to the x-axis:

• This means PR is a horizontal segment. Therefore, P and R must have the same y-coordinate. Let $P = (x_P, y_P)$ and $R = (x_R, y_P)$.
• Since PQ must be perpendicular to PR (the horizontal segment PR), PQ must be a vertical segment. Therefore, P and Q must have the same x-coordinate. Let $Q = (x_P, y_Q)$.
• Also, for a valid triangle, the vertices must be distinct. This means $x_R \ne x_P$ and $y_Q \ne y_P$.

Solution Steps

Step 1: Determine the number of choices for vertex P.

P has coordinates $(x_P, y_P)$.

• Number of choices for $x_P$: 10 (any integer from -4 to 5).
• Number of choices for $y_P$: 11 (any integer from 6 to 16).
• Total number of possible points for P = (Choices for $x_P$) $\times$ (Choices for $y_P$) = $10 \times 11 = 110$.

Step 2: Determine the number of choices for vertex R, given P.

R has coordinates $(x_R, y_P)$. R shares the same y-coordinate as P ($y_P$).

• The x-coordinate $x_R$ must be different from $x_P$.
• Number of choices for $x_R$: Total x-choices minus 1 (excluding $x_P$) = $10 - 1 = 9$.
• The y-coordinate $y_R$ is fixed ($y_R = y_P$).
• Number of choices for R for a fixed P = 9.

Step 3: Determine the number of choices for vertex Q, given P.

Q has coordinates $(x_P, y_Q)$. Q shares the same x-coordinate as P ($x_P$).

• The y-coordinate $y_Q$ must be different from $y_P$.
• Number of choices for $y_Q$: Total y-choices minus 1 (excluding $y_P$) = $11 - 1 = 10$.
• The x-coordinate $x_Q$ is fixed ($x_Q = x_P$).
• Number of choices for Q for a fixed P = 10.

Step 4: Calculate the total number of different triangles.

The total number of triangles is the product of the number of choices for P, the number of choices for R (given P), and the number of choices for Q (given P).

Total Triangles = (Number of choices for P) $\times$ (Number of choices for R) $\times$ (Number of choices for Q)

Total Triangles = $110 \times 9 \times 10 = 9900$.

Alternatively, we can think of choosing the coordinates directly:

Total Triangles = (Choices for $x_P$) $\times$ (Choices for $y_P$) $\times$ (Choices for $x_R \ne x_P$) $\times$ (Choices for $y_Q \ne y_P$)

Total Triangles = $10 \times 11 \times 9 \times 10 = 9900$.

Conclusion

There are 9,900 different right triangles that can be constructed with the given properties and constraints.